00:01
So for the proof here, we first assume that x is going to be completely regular.
00:07
So that's the forward direction.
00:09
And then we can say that for any closed set c, which is going to be a subset of x, and for any point x, and so for all x not in c, there's going to exist, so there exists, a continuous function f sub x c such that x maps to 0, 1 such that f sub x c of x is equal to 0 and f sub x c c is equal to the set containing 1.
01:01
And then we can let f be a collection of all the functions, and we can define a map, let's say a map phi, which is going to be mapping x to the set 0, 1 to the f.
01:14
By we define this by phi of x which is equal to f of x since that f is an element of f and this map is continuous is continuous since f is an element of f are continuous and injective by the distinguishing properties of the functions and the map phi is also going to be a homeomorphism onto its image since x is going to inherit the subspace topology from 0, 1 to the f...